Answer:
steady state temperature =88.7deg C
t=time within 1 deg C of it steady state is 8.31s
Explanation:
A 1 m long wire of diameter 1mm is submerged in an oil bath of temperature 25-degC. The wire has an electrical resistance per unit length of 0.01 Ω/m. If a current of 100 A flows through the wire and the convection coefficient is 500W/m2K, what is the steady state temperature of the wire? From the time the current is applied, how long does it take for the wire to reach a temperature within 1-degC of the steady state value? The density of the wire is 8,000kg/m3, its heat capacity is 500 J/kgK and its thermal condu
The diameter of the wire is known to be=1mm
properties=
The density of the wire is 8,000 kg/m3,
heat capacity is 500 J/kgK
themal conductivity is 20W/m.K
electrical resistance per unit length of 0.01 Ω/m
from lump capavity method
![B_{i} =\frac{hr/2}{k}](https://tex.z-dn.net/?f=B_%7Bi%7D%20%3D%5Cfrac%7Bhr%2F2%7D%7Bk%7D)
500*(2.5*10^-4)/20
0.006<0.1
we know also, to find steady state temperature
Dh(T-Tinf)=![I^{2} R_{e}](https://tex.z-dn.net/?f=I%5E%7B2%7D%20R_%7Be%7D)
make T the subject of the equation , we have
T=25+![\frac{100^2*0.01}{\pi*0.001*500 }](https://tex.z-dn.net/?f=%5Cfrac%7B100%5E2%2A0.01%7D%7B%5Cpi%2A0.001%2A500%20%7D)
T=88.7 degC
rate of chnage in temperature
dT/dt=![\frac{I^2*Re}{rho*c*\pi*D^2/4 } -\frac{4h}{rho*c*D} (T-Tinf)](https://tex.z-dn.net/?f=%5Cfrac%7BI%5E2%2ARe%7D%7Brho%2Ac%2A%5Cpi%2AD%5E2%2F4%20%7D%20-%5Cfrac%7B4h%7D%7Brho%2Ac%2AD%7D%20%28T-Tinf%29)
at t=o and integrating both sides![\frac{T-Tinf-(I^2*Re/\pi*Dh) }{Ti-Tinf-(I^2*Re/\pi*Dh } =exp\frac{-4ht}{rho*c*D}](https://tex.z-dn.net/?f=%5Cfrac%7BT-Tinf-%28I%5E2%2ARe%2F%5Cpi%2ADh%29%20%7D%7BTi-Tinf-%28I%5E2%2ARe%2F%5Cpi%2ADh%20%7D%20%3Dexp%5Cfrac%7B-4ht%7D%7Brho%2Ac%2AD%7D)
we have
![\frac{87.7-25-63.7}{25-25-63.7} =exp\frac{4*500t}{8000*500*0.001}](https://tex.z-dn.net/?f=%5Cfrac%7B87.7-25-63.7%7D%7B25-25-63.7%7D%20%3Dexp%5Cfrac%7B4%2A500t%7D%7B8000%2A500%2A0.001%7D)
t=8.31s
steady state temperature =88.7deg C
t=time within 1 degC of it steady stae is 8.31s